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May  2011, 10(3): 917-936. doi: 10.3934/cpaa.2011.10.917

Feedback control via inertial manifolds for nonautonomous evolution equations

Norbert Koksch 1, and  Stefan Siegmund 2,

1. 

Fachrichtung Mathematik, Technische Universitat Dresden, 01062 Dresden, Germany
2. 

Department of Mathematics, Dresden University of Technology, 01062 Dresden

Received  April 2009 Revised  September 2009 Published  December 2010

In this paper we extend a method to control the dynamics of evolution equations by finite dimensional controllers which was suggested by Brunovsky [3] to nonautonomous evolution equations using nonautonomous inertial manifold theory.
Keywords: feed-back control, process., Inertial manifold, nonautonomous dynamical systems.
Mathematics Subject Classification: Primary: 34C30; Secondary: 34D45, 34G20, 35B42, 37L2.
Citation: Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917
References:
[1]

L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dyn. Stab. Syst., 13 (1998), 265-280.  Google Scholar

[2]

L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: doi:10.1016/S0362-546X(97)00569-5.  Google Scholar

[3]

P. Brunovsky, Controlling the dynamics of scalar reaction diffusion equations by finite dimensional controllers, Proc. IFIP-IIASA Conf., Laxenburg/Austria 1989, Lect. Notes Control Inf. Sci. 154, 22-27 (1991).  Google Scholar

[4]

S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317. doi: doi:10.1016/0022-0396(88)90007-1.  Google Scholar

[5]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Differ. Equations, 13 (2001), 355-380. doi: doi:10.1023/A:1016684108862.  Google Scholar

[6]

C. M. Dafermos, An invariance principle for compact processes, J. Differ. Equations, 9 (1971), 239-252; Erratum. Ibid. 10, 179-180.  Google Scholar

[7]

C. M. Dafermos, Semiflows associated with compact and uniform processes, Math. Systems Theory, 8 (1975), 142-149. doi: doi:10.1007/BF01762184.  Google Scholar

[8]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Application," Pitman Research Notes in Mathematics Series, vol. 27, Longman Scientific & Technical, Harlow, 1992.  Google Scholar

[9]

J. Eisenfeld and V. Lakshmikantham, Comparison principle and nonlinear contractions in abstract spaces, J. Math. Anal. Appl., 49 (1975), 504-511. doi: doi:10.1016/0022-247X(75)90193-6.  Google Scholar

[10]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353.  Google Scholar

[11]

C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Differ. Equations, 1 (1989), 199-244. doi: doi:10.1007/BF01047831.  Google Scholar

[12]

C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Varieties inertielles pour l'equation de Kuramoto-Sivashinsky. (Inertial manifolds for the Kuramoto-Sivashinsky equation), C. R. Acad. Sci., Paris, Ser. I, 301 (1985), 285-288.  Google Scholar

[13]

A. Yu. Goritskiĭ and V. V. Chepyzhov, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 485-511; translation from Mat. Sb., 196 (2005), 23-50.  Google Scholar

[14]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, vol. 850, Springer, 1981.  Google Scholar

[15]

N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equations, 14 (2002), 889-941. doi: doi:10.1023/A:1020768711975.  Google Scholar

[16]

M. A. Krasnoselski, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems, the Method of Positive Operators," Heldermann Verlag, Berlin, 1989.  Google Scholar

[17]

Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete and Continuous Dynamical Systems, 5 (1999), 233-268. doi: doi:10.3934/dcds.1999.5.233.  Google Scholar

[18]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995.  Google Scholar

[19]

L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, pp. 161-168, 1987.  Google Scholar

[20]

J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866.  Google Scholar

[21]

A. V. Romanov, Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47. doi: doi:10.1070/IM1994v043n01ABEH001557.  Google Scholar

[22]

R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dyn. Differ. Equations, 15 (2003), 61-86. doi: doi:10.1023/A:1026153311546.  Google Scholar

[23]

H. Sano and N. Kunimatsu, Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems, IMA J. Math. Control Inform., 11 (1994), 75-92. doi: doi:10.1093/imamci/11.1.75.  Google Scholar

[24]

H. Sano and N. Kunimatsu, An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems, J. Math. Anal. Appl., 196 (1995), 18-42. doi: doi:10.1006/jmaa.1995.1396.  Google Scholar

[25]

G. R. Sell and Y. You, Inertial manifolds: The non-self-adjoint case, J. Differ. Equations, 96 (1992), 203-255. doi: doi:10.1016/0022-0396(92)90152-D.  Google Scholar

[26]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer, New York, 2002.  Google Scholar

[27]

S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziaris, Order reduction for nonlinear dynamic models of distributed reacting systems, Journal of Process Control, 10 (2000), 177-184. doi: doi:10.1016/S0959-1524(99)00029-3.  Google Scholar

[28]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Applied Mathematical Sciences, vol. 68, Springer, New York, 1997.  Google Scholar

show all references

References:
[1]

L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dyn. Stab. Syst., 13 (1998), 265-280.  Google Scholar

[2]

L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: doi:10.1016/S0362-546X(97)00569-5.  Google Scholar

[3]

P. Brunovsky, Controlling the dynamics of scalar reaction diffusion equations by finite dimensional controllers, Proc. IFIP-IIASA Conf., Laxenburg/Austria 1989, Lect. Notes Control Inf. Sci. 154, 22-27 (1991).  Google Scholar

[4]

S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317. doi: doi:10.1016/0022-0396(88)90007-1.  Google Scholar

[5]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Differ. Equations, 13 (2001), 355-380. doi: doi:10.1023/A:1016684108862.  Google Scholar

[6]

C. M. Dafermos, An invariance principle for compact processes, J. Differ. Equations, 9 (1971), 239-252; Erratum. Ibid. 10, 179-180.  Google Scholar

[7]

C. M. Dafermos, Semiflows associated with compact and uniform processes, Math. Systems Theory, 8 (1975), 142-149. doi: doi:10.1007/BF01762184.  Google Scholar

[8]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Application," Pitman Research Notes in Mathematics Series, vol. 27, Longman Scientific & Technical, Harlow, 1992.  Google Scholar

[9]

J. Eisenfeld and V. Lakshmikantham, Comparison principle and nonlinear contractions in abstract spaces, J. Math. Anal. Appl., 49 (1975), 504-511. doi: doi:10.1016/0022-247X(75)90193-6.  Google Scholar

[10]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353.  Google Scholar

[11]

C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Differ. Equations, 1 (1989), 199-244. doi: doi:10.1007/BF01047831.  Google Scholar

[12]

C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Varieties inertielles pour l'equation de Kuramoto-Sivashinsky. (Inertial manifolds for the Kuramoto-Sivashinsky equation), C. R. Acad. Sci., Paris, Ser. I, 301 (1985), 285-288.  Google Scholar

[13]

A. Yu. Goritskiĭ and V. V. Chepyzhov, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 485-511; translation from Mat. Sb., 196 (2005), 23-50.  Google Scholar

[14]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, vol. 850, Springer, 1981.  Google Scholar

[15]

N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equations, 14 (2002), 889-941. doi: doi:10.1023/A:1020768711975.  Google Scholar

[16]

M. A. Krasnoselski, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems, the Method of Positive Operators," Heldermann Verlag, Berlin, 1989.  Google Scholar

[17]

Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete and Continuous Dynamical Systems, 5 (1999), 233-268. doi: doi:10.3934/dcds.1999.5.233.  Google Scholar

[18]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995.  Google Scholar

[19]

L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, pp. 161-168, 1987.  Google Scholar

[20]

J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866.  Google Scholar

[21]

A. V. Romanov, Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47. doi: doi:10.1070/IM1994v043n01ABEH001557.  Google Scholar

[22]

R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dyn. Differ. Equations, 15 (2003), 61-86. doi: doi:10.1023/A:1026153311546.  Google Scholar

[23]

H. Sano and N. Kunimatsu, Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems, IMA J. Math. Control Inform., 11 (1994), 75-92. doi: doi:10.1093/imamci/11.1.75.  Google Scholar

[24]

H. Sano and N. Kunimatsu, An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems, J. Math. Anal. Appl., 196 (1995), 18-42. doi: doi:10.1006/jmaa.1995.1396.  Google Scholar

[25]

G. R. Sell and Y. You, Inertial manifolds: The non-self-adjoint case, J. Differ. Equations, 96 (1992), 203-255. doi: doi:10.1016/0022-0396(92)90152-D.  Google Scholar

[26]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer, New York, 2002.  Google Scholar

[27]

S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziaris, Order reduction for nonlinear dynamic models of distributed reacting systems, Journal of Process Control, 10 (2000), 177-184. doi: doi:10.1016/S0959-1524(99)00029-3.  Google Scholar

[28]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Applied Mathematical Sciences, vol. 68, Springer, New York, 1997.  Google Scholar

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